Contents

# Contents

## Idea

The classical hypergeometric series (introduced by Gauss) are solutions of certain ordinary differential equations of second order.

Special cases appear in classical problems of mathematical physics, solutions to the wave equation, Laplace equation or similar are attacked by Fourier method of separation of variables (cf. Legendre polynomial, Hermite polynomial).

## Definition

The hypergeometric series is defined by the formula,

${}_p F_q (a_1,\ldots,a_p; b_1,\ldots, b_q; x) = \sum_{n=0}^\infty \frac{(a_1)_n (a_2)_n\cdots (a_p)_n}{(b_1)_n (b_2)_n\cdots (b_q)_n}\frac{x^n}{n!}$

where $(a)_0 = 1$ and, for $k = 1,2,3,\ldots$

$(a)_k := a (a+1) (a+2) \cdots (a+k-1)$

is the shifted factorial. In fact let $\sum_{n = 0}^\infty c_n$ be any series of complex numbers such that $c_{n+1}/c_n$ is a rational function of $n$. Then we can find $x,p,q,a_1,\ldots,a_p,b_1,\ldots, b_q$ to write

$\frac{c_{n+1}}{c_n} = \frac{(n+a_1)(n+a_2)\cdots (n+a_p) x}{(n+b_1)(n+b_2)\cdots (n+b_q)(n+1)}$

and $\sum c_n = c_0 {}_p F_q(a_1,\ldots,a_p; b_1,\ldots, b_q; x)$.

There are variants like the confluent hypergeometric function (e.g. Bessel functions), $q$-hypergeometric functions and the basic hypergeometric series. The classical orthogonal polynomials appear as special cases for choices of parameters. There is a recent elliptic version due Spiridonov.

There are now modern generalizations to many variables due Aomoto and another variant due Mikhail Kapranov, Israel Gelfand and Andrei Zelevinsky. These multidimensional generalizations express pairings between representations of quantum groups at root of unity and representations of affine Lie algebras, which can be interpreted as pairings between certain kind of homlogy and cohomology on configuration spaces. This has been extensively studied by Varchenko, Terao and others; often in connection to the study of (complements of) arrangements of hyperplanes in $\mathbb{C}^n$. Selberg-type integrals are involved.

• G. E. Andrews, R. Askey, R. Roy, Special functions, Enc. of Math. and its Appl. 71, Cambridge Univ. Press 1999

• G. Gasper, M. Rahman, Basic hypergeometric series (1990)

• I. M. Gelfand, M. M. Kapranov, A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser 1994, 523 pp.

• Ian G. Macdonald, Hypergeometric functions I, 1987 (arxiv/1309.4568)

In relation to the Knizhnik-Zamolodchikov equation and quantum groups:

Online entries/resources on hypergeometric function:

There is also a far reaching elliptic generalization

• V. P. Spiridonov, Classical elliptic hypergeometric functions and their applications, pdf; Aspects of elliptic hypergeometric functions, arxiv/1307.2876